Synthetic Metals, 9 (1984) 265 - 274 265
THE ROLE OF MOBILE ORGANIC RADICALS AND IONS (SOLITONS, POLARONS AND BIPOLARONS) IN THE TRANSPORT PROPERTIES OF DOPED CONJUGATED POLYMERS
J. L. BREDAS*, B. THEMANS** and J. M. ANDRE Laboratoire de Chimie Thdorique Appliqude, Facultds Universitaires Notre-Dame de la Paix, B-5000 Namur (Belgium) R. R. CHANCE Allied Corporation, Corporate Research Center, Morristown, NJ 07960 (U.S.A.) R. SILBEY Department of Chemistry and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.)
Abstract
We discuss the formation of charged defects such as solitons, polarons and bipolarons, in doped conjugated polymers. We present the results of ab initio quality calculations on the modifications of geometric and electronic structures occurring upon doping. A transport model for spinless conduction through bipolarons applicable to doped polymers with or without degenerate ground state is described.
1. Introduction
During the last few years, the discovery of doped conjugated polymers with high conductivities has generated substantial research interest among chemists and physicists alike, as exemplified in these Proceedings. Besides the potential technological applications of conducting polymers, a very interesting aspect of this field is that a new physics is developing which is significantly different from the physics of doped inorganic semiconductors such as Si, Ge or GaAs. In these introductory remarks, we will point out some of these differences. In inorganic semiconductors, the atoms are connected by covalent bonds in all three dimensions. Carrier mobilities are quite high, so that reasonable conductivities are achieved at fairly low doping levels, of the order of ppm. Doping is usually substitutional and generally does not lead to
*Chercheur Qualifi~ of the Belgian National Science Foundation (FNRS). **Titulaire d'une bourse de sp~cialisation de l'Institut pour l'Encouragement de la Recherche Scientifique dans l'Industrie et l'Agriculture (IRSIA).
0379-6779/84/$ ~ 00 © Elsevier Sequoia/Printed in The Netherlands 266 major disruption of the lattice. Doping of inorganic semiconductors results in the appearance in the gap of electronic levels that can be thermally populated at room temperature. Rigid-band models give a good description of the physics of these systems. In organic polymers, the interactions are highly anisotropic since atoms are linked by strong covalent bonds along the polymer chains, whereas inter- actions between chains are much weaker, usually of the Van der Waals type; this anisotropy opens the door to possible collective instabilities, typical of quasi-one
2. Band structure evolution upon doping of conjugated polymers
So far, polyacetylene (PA) has been the focus of most experimental and theoretical work [1]. Trans-PA is unique among studied systems in possess- ing a degenerate ground state, i.e., two geometric structures having exactly the same total energy. 267
A B These two structures (A and B} differ from one another by the exchange of the alternating single and double carbon-carbon bonds along the chain. This degeneracy leads to the possibility of formation of so-called soliton excita- tions on the chains; in trans-PA, the soliton denotes a one-dimensional domain wall marking the separation and the gradual passage between the two ground-state structures along one chain [ 2, 3 ].
i I i • I , i i B neutral soliton
In principle, this domain wall can propagate freely with a very small kinetic mass, of the order of a few times the free-electron mass. A very im- portant feature is that the presence of a soliton provokes the appearance of a localized electronic level at midgap, i.e., ~0.7 eV above the valence band (VB) edge (Fig. 1). The energy of creation of a soliton is calculated to be about 0.45 eV [2]. Solitons have peculiar charge-spin relationships since neutral solitons have spin 1/2 whereas charged solitons (although carrying a single charge) have no spin. In trans-PA, a neutral soliton (radical) is expected to be present on each chain having an odd number of carbon atoms, this accounts for the concentration of about one spin per 3000 carbon atoms observed in ESR experiments [4]. At low doping levels, charges transferred on the chains can be accom- modated in two ways: (1) as a charged soliton when an existing neutral soliton is ionized; charged solitons are to be viewed simply as polymeric anions (n I IIII ÷ + Fig. 1. Band structure for a polyacetylene chain containing a soliton: left, neutral soliton; middle, positively charged soliton; right, negatively charged soliton. 268 polaron [5-8]. Polarons have spin 1/2 and can also be described as an interacting charged soliton-neutral antisoliton pair. It has been shown [8] that two adjacent polarons recombine (pair their spins) and leave two charged solitons on the chain. As a result, for doping levels of the order of a few per cent, only charged solitons are present on the chains. The corre- sponding picture for the electronic structure is: (1) a full valence band; (2) soliton states around midgap which are fully occupied (n H H The quinoid form has a higher total energy than the aromatic form; the dif- ference per ring is of the order of 0.4 eV [13, 14]. Therefore, it is not pos- sible to have soliton excitations in these systems. The quinoid structure has a lower ionization potential and a larger elec- tron affinity than the aromatic structure (and, as a result, a smaller band- gap). This explains why upon doping the presence of a charge on the chain can provoke a local geometry relaxation from the aromatic structure towards the quinoid structure. The formation of such a charged defect actually occurs when the lowering in ionization energy due to the presence of a 269 quinoid segment more than compensates for the increase in ~ + o energy required to form that quinoid segment. We have performed calculations on PPP [8] and PPy [15], using Huckel theory with sigma compressibility, i.e., the quantum-chemical equivalent of the Su-Schrieffer-Heeger Hamiltonian as applied to PA [2]. We find that, at low doping levels, radical-ions (polarons) are formed, associated with a quinoid segment extending over 4 to 5 rings. The presence of a polaron introduces two states in the gap, a 'bonding' state slightly above the VB edge and an 'antibonding' state slightly below the CB edge. The calculations indicate that two adjacent polarons are unstable with respect to the pairing of their spins and the formation of a bipolaron, i.e., a doubly-charged defect having no spin [8, 15]. Note that in contrast to trans-PA, since solitons are not possible in these systems, the charges cannot move separately but are correlated in pairs. The geometrical defect associated with a bipolaron has a width similar to that of the polaron. However, the bond-length modifications are larger so that the two states in the gap are pushed further away from the band edges. The absence of Pauli susceptibility in the highly conducting regime of PPP [16] and PPy [17] is consistent with our suggestion [8, 15] that the bipolarons are the corresponding spinless charge carriers. We have calculated at the ab initio level the bond-length modifications appearing in PPP [18], PPy [19] and PT [14] when a bipolaron is formed and the band-structure evolution at high doping levels. Using Hartree-Fock ab initio techniques, we optimize fully the geometry of a large oligomer (quaterphenyl, quaterpyrrole or quaterthiophene); the geometric optimiza- tion is carried out first in the undoped state, then in the doped state by adding two Li or Na atoms in order to simulate the formation of a bipolaron on the chain. Typical bond-length modifications are given in Table 1. Taking the geometries obtained in that way for the charged defects and using the ab initio quality Valence Effective Hamiltonian (VEH) technique [ 20, 21], we calculate the electronic structure of polymer chains: ( 1 ) in the undoped state; (2) containing one bipolaron; (3) doped at the maximum level experimentally achieved, i.e., 50% (on a per monomer basis) for PPP and 33% for PPy and PT; (4) for a hypothetical 100% doping level. The band structure evolution for PPP is presented in Fig. 2. In the undoped state (Fig. 2(a)), the bandgap is calculated to be 3.5 eV, in good agreement with experiment [22]. The presence of a bipoiaron provokes the appearance of two localized electronic states in the gap, 0.7 eV away from the band edges (Fig. 2(b)). In case of n-(p-)type doping, these two states are fully occupied (empty). At high doping levels, the bipolaron states overlap and broaden to form bipolaron bands. At 50% doping level, usually achieved with alkali metal ions, the two bipolaron bands are about 0.5 eV wide (Fig. 2(c)). This picture is in agreement with recent electron energy loss data [23]. For a hypothetical 100% doping level, the lower bipolaron band merges with the VB and the upper one with the CB (Fig. 2(d)). A traditional conduction mechanism would then occur due to the un- filled character of either the VB (p-doping) or the CB (n-doping), in much 270 TABLE 1 Results of Hartree-Fock ab initio geometry optimizations on tetramers of polypara- phenylene, polypyrrole and polythiophene, undoped and doped with two sodium atoms. Bond lengths are given in A Undoped Doped Polyparaphenylene r(C--C) 1.384 - 1.393 r(C=C) 1.352 r(C--C) 1.462 r(inter-ring) 1.507 1.371 Polypyrrole r(C--N) 1.385 1.428 r(C--C) 1.424 1.380 r(C=C) 1.363 1.437 r(inter-ring) 1.474 1.357 Polythiophene r(C--S) 1.721 1.776 r(C---C) 1.444 1.370 r(C=C) 1.346 1.432 r(inter-ring) 1.480 1.362 3.5eV 1.0eV ® ® ® ® Fig. 2. Band-structure evolution upon (n-)doping of polyparaphenylene: (a) undoped; (b) intermediate doping level: non-interacting bipolarons present on the chain; (c) per monomer 50% doping level; (d) per monomer 100% doping level. the same way as what happens in PA above 7% doping. Here, however, note that the original bandgap would not close (but rather decreases to the band- gap value in the purely quinoid form). The band structure evolution calculated for PPy is very similar to that presented for PPP, except that the locations of the bipolaron states in the gap are asymmetric with respect to the gap centre [19]. The bonding state is located about 0.45 eV above the VB edge and the antibonding state, 0.9 eV below the CB edge. This asymmetric location seems to be confirmed exper- imentally [15]. It is due to the fact that in PPy the atoms contributing to the highest occupied molecular orbital (HOMO) are quite different from those contributing to the lowest unoccupied MO (LUMO); e.g., the HOMO has no contributions from the nitrogen lr orbitals whereas the LUMO does. 271 In PT the bandgap is calculated to be significantly smaller, 2.2 eV, in agreement with experiment [24]. The bipolaron states appear 0.6 and 0.7 eV away from the band edges (the asymmetry is thus smaller than in PPy, although the HOMO and LUMO characteristics are the same as in PPy). The bipolaron bands formed at 33% doping level have widths of the order of 0.3 eV. So far, PPy is the highly conducting polymer where experimental and theoretical evidence for the presence and role of spinless bipolarons in the transport process is most conclusive. Now that a clean chemical synthesis has been developed for PT [24], it will be very interesting to look for similar evidence in this compound. 3. Carrier transport in polyacetylene and polypyrrole A question of fundamental importance in the conducting polymers area is how the charged species (solitons, bipolarons) we have described in the previous section can migrate through the polymer/dopant array. As pointed out above, the question is made very interesting by the observation of spin- less transport, i.e., a spin concentration that is far too low to account for the observed conductivities [9, 10, 16, 17]. In this section, we discuss a model based on bipolarons which could account for truly spinless transport at intermediate and high doping levels in trans-PA and non-degenerate ground state systems. A detailed presentation of this model is given elsewhere [25]. In trans-PA, a number of authors [2, 9] have suggested that charged solitons are the charge carriers in the spinless conductivity regime (up to about 7% doping level). However, they fail to consider how these species can get from one chain to another, it being well-established that interchain transport must be easy for efficient carrier migration. Actually, interchain conduction is likely to be the rate-limiting step in all conducting polymer systems. In Fig. 3, we consider an infinite PA chain containing one charged soliton and a neighbouring chain without defect. The soliton cannot easily hop to the next chain since an infinitely large activation barrier is required in the reorganization of bond lengths that would ensue from such a jump. The soliton mechanism is also unattractive for finite chains, since the soliton can- not hop isoenergeticany in that case. The model developed by Kivelson [26] gets around those problems by requiring a neutral soliton on the next chain. However, this model is only applicable at low doping levels [26]. The above problems are eliminated by a model in which there are two charged solitons on an infinite chain. The soliton pair, that can be referred to as a bipolaron, can hop to the next chain isoenergetically and with a reasonable activation barrier (Fig. 3). The hopping probability for two charged solitons separated by x carbon atoms, P(x), is constructed as follows: P(x) = F(x) Qa(x) Qb(x) 272 Polyacetylene Soliton hopping Bipolaron hopping * ° • ~ ) ° • ° Polypyrrole Bipolaron hopping " ° ° ~ ° ° ° ) H H N N H Fig. 3. Interehain transport for solitons and bipolarons in trans-polyaeetylene and bi- polarons in polypyrrole. where F{x) is the Franck-Condon factor (or square of the vibrational over- lap) for the jump; Qa(x) is the probability of finding a second charged soliton x carbon atoms from the first on a single chain; Qb(x) is the probabil- ity of finding an unoccupied site on the next chain to receive the x-length soliton pair. F(x) is expected to decrease rapidly as x increases. Qa(x) is expected to be peaked at roughly 1/C (C is the concentration in solitons, taken to be the dopant concentration) due to the coulombic repulsion be- tween solitons of like charge. Qb(x) is obtained from Qa(x) and leads to saturation of the spinless contribution to the conductivity at high C. We have modelled this process and calculated conductivity versus C curves that are very similar to those obtained experimentally. We now discuss the qualitative aspects of the results of the model. At low C, there are few soliton pairs, 'bipolarons', of small enough x for transport to take place due to the F(x) term. Indeed, F(x) at the peak in the Qa(x) distribution is small. As C increases, the spinless contribution to the conductivity increases superlinearly since the peak in Qa(x) shifts to lower x values where F(x) gets very large. At still larger C values, the spinless contribution levels off and eventually decreases due to the small value of the Qb(x) term. The reason is 273 that few sites remain available on the next chain to accept the bipolaron. The net result of the model is an S-shaped curve for the spinless contribution to the conductivity. For PPP, PPy or PT, the situation is different since the two charged defects which make up the bipolaron are bound. The bipolaron is thus a well~efined entity of limited spatial extent, so that the F(x) and Qa(x) terms become constant. Therefore, the spinless contribution to the conduc- tivity rises in proportion to C, eventually saturating and decreasing when the Qb(x) term becomes small. This qualitative behaviour is consistent with the experimental data in PPy [27], although no detailed conductivity versus doping level curve has been reported so far for any of these systems. Bipolaron transport offers a reasonable explanation for spinless transport in PA, PPP, PPy and PT. It should be emphasized, however, that we have made no explicit consideration of the effect of the dopant ion array on the transport process. 4. Conclusions In summary, we offer the following qualitative picture of the doping and transport process of general applicability to conducting polymers. The dopant (which we take to be an acceptor) first ionizes the chain to produce a polaron (radical-cation). The polaron is pinned to the counter-ion and does not contribute significantly to conductivity. As more dopant is added, the chain is ionized further and a higher concentration of polarons is formed. Also, the polarons can be further ionized to produce bipolarons (dications) since the polaron can be more easily ionized than the polymer chain. As the polaron concentration gets higher, adjacent polarons interact to produce bi- polarons, which are uncorrelated charged solitons in the PA case and cor- related dications in the other cases. The bipolarons transport charge via inter- chain hopping and are responsible for the observed spinless conductivity. The corresponding picture for the electronic structure reveals the presence of a full valence band, an empty conduction band and wide bands within the gap due to solitons in the PA case and bipolarons in the other cases. In cases where still higher doping levels can be achieved, the bands in the gap merge with the VB and CB; at this point, conventional conductivity involving carriers with spin sets in, as for PA above 7% doping. It should be stressed that this description applies equally well for systems with or without a de- generate ground state. Acknowledgements We thank the organizers of the Los Alamos Symposium on Synthetic Metals for their invitation to present this paper. JLB is grateful to the Belgian National Science Foundation (FNRS) for continuous support. 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